Gauge theory geography: charting a path between semiclassical islands

We study two semiclassical limits of \(SU(2)\) Yang-Mills theory on a spatial torus with a 't Hooft twist: the ``femtouniverse,'' where all \(\mathbb{T}^3\) directions are small, and deformed Yang-Mills theory on \(\mathbb{T}^2 \times \mathbb{S}^1\), with small \(\mathbb{S}^1\) and large or infinite \(\mathbb{T}^2\). Carefully defining the symmetries, we show that the classical ground states, while different, have the same transformation properties under the 1-form center symmetry and parity. We argue that this is behind the identical multi-branch \(\theta\)-dependent vacuum structure of these theories. We then calculate the one-loop potential for the \(\mathbb{S}^1\)-holonomy in the presence of twists on \(\mathbb{T}^2\). We use it to study the quantum stability of the semiclassical ground states in gauge theories with massive or massless adjoint fermions on spatial \(\mathbb{T}^2 \times \mathbb{S}^1\), with a twist in the \(\mathbb{T}^2\). The results point towards some interesting features worthy of further study.